Velocity Scheduled Flow Matching

Flow matching trains a neural network to regress the conditional velocity along a linear interpolant between noise and data, and the number of network evaluations~(NFE) sets the cost of sampling. The straight-line interpolant carries an implicit choice: the sample moves at constant speed throughout the trajectory. We relax this choice and introduce Velocity Scheduled Flow Matching~(VSFM), which replaces the conditional target x1 - x0 with v(t)(x1 - x0) for any nonnegative profile v:[0,1]→ℝ≥ 0 satisfying int01 v,dt = 1. We study six polynomial profiles drawn from motion planning. The first use of VSFM is at inference time: a pretrained linear flow-matching model can be sampled under any admissible profile by integrating its ODE on a non-uniform τ-schedule, with no retraining and no additional computation; on CIFAR-10 this lowers FID by up to 19.8%. Training from scratch under a braking profile gives a further reduction of 17.4% at 4~NFE. Both gains follow from the local truncation error of the Euler integrator on the induced grid.